Month: August 2012

It’s been a remarkably swift summer. It feels like just yesterday that I was sitting in exactly the same spot I am now (this lovely little café) writing end-of-year reports… Quite a lot has happened in the last two and a half months (including an entire career with the Museum of Mathematics – which was very interesting and educational for me – maybe I’ll write about this in the future). But, now, it’s time to think about the coming year. Here I go!

One of the reasons I’m thankful I got my first teaching job at Saint Ann’s (apart from the stellar faculty and intellectual atmosphere that fills even a third grade classroom) is that I got to make significant choices about the content I teach in my classes. I not only had to decide how to teach, for example, multiplication of fractions to my sixth graders, but also when to teach it and – most importantly – whether or not to teach it!

Now, of course, it’s strongly encouraged to teach fraction multiplication to sixth graders. But, especially in my first year, guidelines were weak and sparse. And they’re just that – guidelines. They aren’t requirements. As math teachers, we’re trusted to know what’s best for our students’ math education. And if I’d deemed it unwise to teach fraction multiplication to my sixth graders, my judgment would have been trusted (probably).

This freedom and trust prompted me to think very carefully about the actual mathematics I was teaching my students. With an entire curriculum to devise, I had to decide what was valuable for my students to learn and do. I’ve really enjoyed doing this and have learned a great deal about mathematics in the process. I really don’t think I could have learned these things elsewhere.

But this meant that I’ve spent much of the past three years thinking about content, not about classroom culture. Now, content informs classroom culture. And I’ve done some thinking about classroom culture out of necessity. But, here in my fourth year as a math teacher, with two filing cabinet drawers full of content to my name, it’s about time I did some concentrated thinking about what I want to go on in my classroom.

In an attempt to start thinking about that, I decided to first describe what actually does go on in my classroom. I’ve never thought about this in great depth. When I start a unit, I think very carefully about what I want to teach and how I want to present it. I structure the development of ideas and plan out the investigations that my students do. But I don’t think nearly as carefully about who says what and when, the nature and structure of assignments, when and how I’ll give feedback, and how to make sure that everyone is engaged in a way that feels good to them. When I started thinking about this, I realized that last year I sort of de facto developed a unit structure that I actually really like. There are ways I think this structure can be improved, however. But I can’t know how to tinker with the structure without an honest picture of what it’s really like!

So, I did this close study of my unit structure and classroom culture. I first identified some questions that I’m looking to answer as I describe and analyze my classroom culture. Then, I listed stages that my units generally move through. I then described in detail how those stages go, using the questions. Finally, I developed additional questions to help me focus on areas for improvement within the stages.

This process was really quite helpful to me. It was surprising to me to recognize that I really do have a unit structure that I follow in most all of my classes. It was eye-opening to analyze this structure for the values that I hold. My unit structure does channel many of my values – emphasizing investigation and student choice, making a student-centered classroom, allowing students to move through material in their own time and communicate and evaluate their learning in their own ways, and encouraging student questioning and discovery. But there are ways that I haven’t communicated all of my values and have under and over emphasized some aspects of math education.

If you’re interested, you can read on for the full details of my self-study. It’s quite long. I find it interesting – not sure how interesting it’ll be to anyone else. But go for it, if you’re game.

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Questions:

Who does what and with whom?

Who says what and to whom?

Who’s leading and following, and under what circumstances?

How can I do a better job of making all voices heard in a way that feels best to them?

What do I emphasize – intentionally and de facto? How are my goals made clear?

Do I emphasize all of the things that I want to emphasize?

How do I set standards, provide guidance, and give feedback?

Where can I integrate more feedback into this structure? What form should that feedback take?

Aspects of class:

– Questioning

– Hypothesizing

– Investigation

– Discovery

– Formulation

– Communication

– Practice

– Performance

– Cycle back to questioning

How my classes/units usually go:

1. Start with a question. The question is sometimes motivated independently of me by a student’s curiosity. Sometimes I motivate it by providing problems on a new topic or altering something we’re already studying. At some point, either after students have had time to be stumped, to see a variety of outcomes, to sniff a whiff of a pattern, or to somehow have their curiosity aroused, we pose a question or more. This usually happens during a class discussion.

How can this be done less under my direction?

How can questions be solicited more organically?

2. Make some hypotheses. If I’ve motivated the question with problems, then the students have some experience with the curious material already. If it’s independently motivated, the student who raised the question will share some thoughts that he or she has had. (This usually happens during a class discussion.) With this experience, we predict possible answers to the questions we’ve raised. This is phrased as, “How could this even remotely plausibly turn out?” to encourage all possibilities, even those that may not be true. We evaluate them as best we can, given our knowledge.

How can this be done to encourage broader participation?

How can I encourage students to think more carefully about how the situation might turn out before we test it?

3. Investigate and discover. Sometimes the students develop their own ways to investigate the questions. I field these suggestions, provide some of my own, and, as a class, we decide how to proceed. Sometimes, though, I skip this step and give them an investigation of my own design. In any case, I come up with the problems for use during investigation. I most frequently will make a worksheet. Then, the students work through these problems. They usually each have their own copy of the same thing and are encouraged to work together. Often, there’s a way for them to keep track of their findings. Sometimes I just encourage them to look for answers to the questions as they work. Sometimes they can share their findings with the whole class as they find them. Sometimes they’re encouraged to keep it to themselves until everyone has finished. This almost always takes place during class; when it’s homework, it is less successful. Class during this stage usually looks like: students at desks or walking around the room, working and talking; teacher circulating the room, often being called over by excited or confused students for help.

How can the investigation be structured and developed more by students than by me? How do I teach that?

How can this be done outside of the classroom? (And do I really want that?)

How can I design investigations so that each student’s discoveries are more important?

How can I better keep track of and evaluate students’ investigation skills and abilities?

4. Discover and formulate. Often, at the end of the worksheet will be a series of problems that you need to know the answers to the questions we’ve posed to solve – they’re really complicated in some way that makes them virtually impossible without the “trick,” but very simple with the “trick.” There will also be questions that prompt a written description of what’s going on. This is if I have a set outcome that I want everyone to reach. If I don’t have a set outcome (or, sometimes, even if I do), students will be assigned to write a description of their investigation process and what they found. Sometimes this is prompted with questions that tug out the discovery. The emphasis is on writing words that describe the process and discovery – explanation is encouraged, but explicitly not expected. And written words are valued over other means of explication. This is typically homework, but sometimes takes place during class. It’s also usually an individual task.

How can I incorporate methods of formulation and communication that aren’t biased for the written and verbal?

How can I better give feedback on the individual formulation that students do?

How can I help students set goals for this part of the process and improve at it?

5. Communicate personal discoveries and formulations, and formulate the discoveries and ideas into theories, rules, and methods that can be communicated to each other. This usually takes place during a class discussion, when students share their discoveries with the whole class. Sometimes they share their discoveries and writing with each other in small groups first and are encouraged to come to consensus where there’s disagreement. Eventually, though, everything is settled in a class discussion. As students share, I prompt more formal formulation of theories, rules, and methods – usually calling them by the name of the kid or kids who suggested them. I write them on the board as clear steps and in straight-forward language. Students are encouraged to record this. Often multiple theories and methods about the same thing are formulated.

How can the decision-making and consensus-reaching be done less during a class discussion led by me and more among the students themselves?

6. Practice using the theories, rules, and methods. Students are given problems I’ve written where they can practice using what we’ve developed. This sometimes takes place during class, sometimes as homework. The worksheet with practice problems sometimes includes problems at the end that have a twist – it’s not clear how to apply the rule or method, applying the rule or method is much more difficult than on other problems or doesn’t apply at all. This encourages students to pose new questions, and the cycle begins – sometimes on a smaller scale, with just altering the original method or theory, but sometimes on a large scale so that the process begins again.

7. Demonstrate, perform, and share what you’ve learned, what you’ve done, and what you can do. Some of this performance takes place when the students are sharing their discoveries – but that’s less a performance of a final product than of the working process. Also, not everyone will necessarily share something at that stage, and some students’ ideas are clearly used more than others in the work that follows. The only performance that everyone does is taking skills challenges that allow students to demonstrate how they can use the rules and methods to solve problems like the ones we started posing questions about.

What can I do to help students perform things other than their mastery of skills? Other things to perform are:

– Thorough understanding of the theories and methods in an abstract, rather than practical and application-based, way

– Ability to apply and tailor theories and methods to different problems

– Ability to communicate understanding of the theories and methods

– Ability to teach what we’ve discovered to others

– Facility with this whole process

– Personal discoveries they made during this process

– Ability to come up with new questions to investigate

How important is it that the same type of performance be done by all students?

Are certain kinds of performance “better” than others?

What is the goal of performance for the students?

What purpose does performance serve for my own evaluation of their learning?

8. Cycle back. When students have new questions and ideas, they are encouraged to present about them. They come to the board and explain their new findings to their classmates. And then we start over again.